In the comics, the original Cosmic Cube (aka Tesseract) was created by AIM's scientists and used by the Red Skull in his various schemes. Hawkeye recruited a group of rogue scientists to work on the Tesseract in The Avengers, and they could well have been members of AIM. (HYDRA is a less likely possibility because the group Clint and Selvig were working with were from a variety of ethnicities, which would be anathema to the Nazi affiliated villains.)

With that background in mind, it would make perfect sense if AIM succeeded in creating (or recreating) the Tesseract in CA: TWS.

I don't think that, sixty years later, HYDRA would still care about Nazi racial ideology. Sure, no explicit evidence, but its not like Red Skull overly cared about such, himself.